DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS

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(1)DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS Yann Chaubet, Nguyen Viet Dang. To cite this version: Yann Chaubet, Nguyen Viet Dang. DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS. 2019. �hal-02371379�. HAL Id: hal-02371379 https://hal.archives-ouvertes.fr/hal-02371379 Preprint submitted on 19 Nov 2019. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS YANN CHAUBET AND NGUYEN VIET DANG. Abstract. We introduce a new object, called dynamical torsion, which extends the potentially ill-defined value at 0 of the Ruelle zeta function ζ of a contact Anosov flow twisted by an acyclic representation of the fundamental group. The dynamical torsion depends analytically on the representation and is invariant under deformations among contact Anosov flows. Moreover, we show that the ratio between this torsion and the refined combinatorial torsion of Turaev, for an appropriate choice of Euler structure, is locally constant on the space of acyclic representations. In particular, for contact Anosov flows path connected to a geodesic flow of a hyperbolic manifold among contact Anosov flows, we relate the leading term of the Laurent expansion of ζ at the origin, the Reidemeister torsion and the torsions of the finite dimensional complexes of the generalized resonant states of both flows for the resonance 0. This extends previous work of [DGRS18] on the Fried conjecture near geodesic flows of hyperbolic 3–manifolds, to hyperbolic manifolds of any odd dimensions.. 1. Introduction Let M be a closed odd dimensional manifold and (E, ∇) be a flat vector bundle over M . The parallel transport of the connection ∇ induces a conjugacy class of representation ρ ∈ Hom(π1 (M ), GL(Cd )). Moreover, ∇ defines a differential on the complex Ω• (M, E) of E-valued differential forms on M and thus cohomology groups H • (M, ∇) = H • (M, ρ) (note that we use the notation ∇ also for the twisted differential induced by ∇ whereas it can be denoted by d∇ in other references). We will say that ∇ (or ρ) is acyclic if those cohomology groups are trivial. If ρ is unitary (or equivalently, if there exists a hermitian structure on E preserved by ∇) and acyclic, Reidemeister [Rei35] introduced a combinatorial invariant τR (ρ) of the pair (M, ρ), the so-called Franz-Reidemeister torsion (or R-torsion), which is a positive number. This allowed him to classify lens spaces in dimension 3; this result was then extended in higher dimension by Franz [Fra35] and De Rham [dR36]. Later, Ray-Singer [RS71] introduced another invariant τRS (ρ), the analytic torsion, defined via the derivative at 0 of the spectral zeta function of the Laplacian given by the Hermitian metric on E and some Riemannian metric on M . They conjectured the equality of the analytic and Reidemeister torsions. This conjecture was proved independently by Cheeger [Che79] and M¨ uller [M¨ ul78], assuming only that ρ is unitary (both R-torsion and analytic torsion have a natural extension if ρ is unitary and not acyclic). The Cheeger-M¨ uller theorem was extended to unimodular flat vector bundles by M¨ uller [Mul93] and to arbitrary flat vector bundles by Bismut-Zhang [BZ92]. To remove indeterminacies arising in the definition of the combinatorial torsion, Turaev [Tur86, Tur90, Tur97] introduced in the acyclic case a refined version of the combinatorial 1.

(3) 2. Y. CHAUBET AND N.V. DANG. R-torsion, the refined combinatorial torsion. It is a complex number τe,o (ρ) which depends on additional combinatorial data, namely an Euler structure e and a cohomological orientation o of M . We refer the reader to subsection 8.2 for precise definitions. Later, Farber-Turaev [FT00] extended this object to non-acyclic representations. In this case, τe,o (ρ) is an element of the determinant line of cohomology det H • (M, ρ). Motivated by the work of Turaev, but from the analytic side, Braverman-Kappeler [BK07c, BK+ 08, BK07b] introduced a refined version of the Ray-Singer analytic torsion called refined analytic torsion τan (ρ). It is complex valued in the acyclic case. Their construction heavily relies on the existence of a chirality operator Γg , that is, Γg : Ω• (M, E) → Ωn−• (M, E),. Γ2g = Id,. which is a renormalized version of the Hodge star operator associated to some metric g. They showed that the ratio τan (ρ) ρ 7→ τe,o (ρ) is a holomorphic function on the representation variety given by an explicit local expression, up to a local constant of modulus one. This result is an extension of the Cheeger-M¨ uller theorem. Simultaneously, Burghelea-Haller [BH07] introduced a complex valued analytic torsion, which is closely related to the refined analytic torsion [BK07a] when it is defined; see [H+ 07] for comparison theorems. In the context of hyperbolic dynamical systems, Fried [Fri87] was interested in the link between the R-torsion and the Ruelle zeta function of an Anosov flow X which is defined by Y −s`(γ) , Re(s) 0, ζX,ρ (s) = det 1 − εγ ρ([γ])e # γ∈GX. # where GX is the set of primitive closed orbits of X, `(γ) is the period of γ and εγ = 1 if the stable bundle of γ is orientable and εγ = −1 otherwise. Using Selberg’s trace formula, Fried could relate the behavior of ζX,ρ (s) near s = 0 with τR , as follows.. Theorem 1 (Fried [Fri86]). Let M = SZ be the unit tangent bundle of some closed oriented hyperbolic manifold Z, and denote by X its geodesic vector field on M . Assume that ρ : π1 (M ) → O(d) is an acyclic and unitary representation. Then ζX,ρ extends meromorphically to C. Moreover, it is holomorphic near s = 0 and r. |ζX,ρ (0)|(−1) = τR (ρ),. (1.1). where 2r + 1 = dim M , and τR (ρ) is the Reidemeister torsion of (M, ρ). In [Fri87], Fried conjectured that the same holds true for negatively curved locally symmetric spaces. This was proved by Moscovici-Stanton [MS91], Shen [She17]. For analytic Anosov flows, the meromorphic continuation of ζX,ρ was proved by Rugh [Rug96] in dimension 3 and by Fried [Fri95] in higher dimensions. Then Sanchez-Morgado [SM93, SM96] proved in dimension 3 that if ρ is acyclic, unitary, and satisfies that ρ([γ]) − εjγ is invertible for every j ∈ {0, 1} for some closed orbit γ, then (1.1) is true..

(4) DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS. 3. For general smooth Anosov flows, the meromorphic continuation of ζX,ρ was proved by Giuletti-Liverani-Pollicott [GLP13] and alternatively by Dyatlov–Zworski [DZ16], the Axiom A case was treated by Dyatlov–Guillarmou in [DG18]. These results are important byproducts of new functional methods to study hyperbolic flows. All these works rely on the construction of spaces of anisotropic distributions adapted to the dynamics, initiated by Kitaev [Kit99], Blank–Keller–Liverani [BKL02], Baladi [Bal05, Bal18], Baladi–Tsujii [BT07], Gouëzel-Liverani [GL06], Liverani [Liv05], Butterley-Liverani [BL07, BL13], and many others where we refer to the recent book [Bal18] for precise references. These spaces allow to define a suitable notion of spectrum for the operator L∇ X = ∇ιX + ιX ∇, where ι is the in• terior product, acting on Ω (M, E). This spectrum is the set of so-called Pollicott–Ruelle resonances Res(L∇ X ), which forms a discrete subset of C and contains all zeros and poles of ζX,ρ . Faure–Roy–Sj¨ ostrand [FRS08], Faure–Sjöstrand [FS11] initiated the use of microlocal methods to describe these anisotropic spaces of distributions giving a purely microlocal approach to study Ruelle resonances. Combining this point of view with propagation estimates originating from scattering theory [Vas13, Mel94] allowed the authors of [DZ16] to −1 control the wavefront set of the resolvent L∇ yielding another proof of the analytic X +s continuation of the zeta function. Using this microlocal approach, one could try to address questions relating topology and dynamical zeta functions. Quoting the commentary from Zworski [Zwo18] on Smale’s seminal paper [Sma67], equation (1.1) ”would link dynamical, spectral and topological quantities. [. . . ] In the case of smooth manifolds of variable negative curvature, equation (1.1) remains completely open”. However in [DZ17], the authors were able to prove the following. Theorem 2 (Dyatlov–Zworski). Suppose (Σ, g) is a negatively curved orientable Riemannian surface. Let X denote the associated geodesic vector field on the unitary cotangent bundle M = S ∗ Σ. Then for some c 6= 0, we have as s → 0 ζX,1 (s) = cs|χ(Σ)| (1 + O(s)) ,. (1.2). where 1 is the trivial representation πn1 (S ∗ Σ) → C∗ o and χ(Σ) is the Euler characteristic of # Σ. In particular, the length spectrum `(γ), γ ∈ GX determines the genus. This result was generalized in a recent preprint of Cekić–Paternain [CePa] to volume preserving Anosov flows in dimension 3. In the same spirit and using similar microlocal methods, Guillarmou-Rivière-Shen and the second author [DGRS18] showed Theorem 3 (D–Rivière–Guillarmou–Shen). The map X 7→ ζX,ρ (0) is locally constant on the open set of smooth vector fields which are Anosov and for which 0 is not a Ruelle resonance, that is, 0 ∈ / Res(L∇ X ). By an approximation argument, equation (1.1) holds true for M of dimension 3, if X preserves a smooth volume form, and b1 (M ) 6= 0 or under the same assumption used in [SM96], still assuming ρ unitary and acyclic..

(5) 4. Y. CHAUBET AND N.V. DANG. If 0 ∈ Res(L∇ X ), the results of [DGRS18] no longer apply. In the case where X induces a contact flow, which means X = Xϑ is the Reeb vector field of some contact form ϑ on M , we overcome this difficulty by introducing a dynamical torsion τϑ (ρ) which is defined for any acyclic ρ and which coincides with ζX,ρ (0)±1 if 0 ∈ / Res(L∇ X ). To do so, in the spirit of [BK07c], we use a chirality operator associated to the contact form ϑ, Γϑ : Ω• (M, E) → Ωn−• (M, E),. Γ2ϑ = Id,. cf. §5, analogous to the Hodge star operator associated to a metric. Let us briefly describe 0 the construction of the dynamical torsion. Let C • ⊂ D • (M, E) be the finite dimensional space of Pollicott-Ruelle generalized resonant states of L∇ X for the resonance 0, that is, n o 0 N C • = u ∈ D • (M, E), WF(u) ⊂ Eu∗ , ∃N ∈ N, (L∇ ) u = 0 , X where WF is the H¨ ormander wavefront set, Eu∗ ⊂ T ∗ M is the dual of the stable bundle of X, cf. §4, and D0 (M, E) denotes the space of E-valued currents. Then ∇ induces a differential on the complex C • and a result from [DR17c] implies that the complex (C • , ∇) is acyclic if • we assume that ∇ is. Because Γϑ commutes with L∇ X , it induces a chirality operator on C . Therefore we can compute the torsion τ (C • , Γϑ ) of the finite dimensional complex (C • , ∇) with respect to Γϑ , as described in [BK07c] (see §2). Then we set q. q. q. τϑ (ρ)(−1) = τϑ (∇)(−1) = ±τ (C • , Γϑ )(−1) lim s−m(X,ρ) ζX,ρ (s) ∈ C \ 0, s→0. where the sign ± will be given later, m(X, ρ) is the order of ζX,ρ (s) at s = 0 and q is the dimension of the unstable bundle of X. Note that the order m(X, ρ) ∈ Z is a priori not stable under perturbations of (X, ρ) whereas τϑ (ρ) has interesting invariance properties as we will see below. We denote by Repac (M, d) the set of acyclic representations π1 (M ) → GL(Cd ) and by A ⊂ C ∞ (M, T M ) the space of contact forms on M whose Reeb vector field induces an Anosov flow. This is an open subset of the space of contact forms. For any ϑ ∈ A, we denote by Xϑ its Reeb vector field. The fact that the dynamical torsion τϑ (ρ) is defined even if 0 ∈ Res(L∇ Xϑ ) allows us to compare it with Turaev’s refined torsion. The main result of this paper is the following Theorem 4. Let (M, ϑ) be a contact manifold such that the Reeb vector field of ϑ induces an Anosov flow. Then the following holds true. (1) There exists a canonical Euler structure eϑ associated to ϑ such that for any cohomological orientation o, the map τϑ (ρ) ρ ∈ Repac (M, d) 7−→ τeϑ ,o (ρ) is locally constant on Repac (M, d). Moreover, if dim M = 3 and b1 (M ) 6= 0, this map is of modulus one on the connected components of Repac (M, d) containing an acyclic and unitary representation. (2) The map. τη (ρ). (η, ρ) ∈ A × Repac (M, d) 7−→ det ρ, cs(Xη , Xϑ ) τϑ (ρ).

(6) DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS. 5. is locally constant on A×Repac (M, d), where for any nonvanishing vector fields Y, X, cs(Y, X) ∈ H1 (M, Z) is the Chern-Simons class of the pair (Y, X). The Chern–Simons class cs(Y, X) ∈ H1 (M, Z) measures the obstruction to find a homotopy among non singular vector fields connecting Y and X. In particular, if ϑ and η are connected by some path in A, then cs(Yη , Xϑ ) = 0, and thus τϑ (ρ) = τη (ρ) for any acyclic ρ. We refer the reader to subsection 8.1 for the definition of Chern-Simons classes. This theorem generalizes some results from [DGRS18] in the case of contact flows. Because the dynamical torsion is constructed with the help of the dynamical zeta function ζX,ρ , we deduce from the above theorem some informations about the behavior of ζX,ρ (s) near s = 0. Corollary 5. Let M be a closed odd dimensional manifold. Then for every connected open subsets U ⊂ Rep0 (M, d) and V ⊂ A, there exists a constant C such that for every Anosov contact form ϑ ∈ V and every representation ρ ∈ U, q. ζXϑ ,ρ (s)(−1) = Cs(−1). q m(ρ,X. τeXϑ ,o (ρ). ϑ). τ (C • (ϑ, ρ) , Γϑ ). (1 + O(s)) ,. (1.3). where Xϑ is the Reeb vector field of ϑ, (Eρ , ∇ρ ) is the flat vector bundle over M induced by 0 ρ, C • (ϑ, ρ) ⊂ D • (M, Eρ ) is the space of generalized resonant states for the resonance 0 of ∇ LXϑρ and m(Xϑ , ρ) is the vanishing order of ζXϑ ,ρ (s) at s = 0. Combining these results with Fried’s Theorem 1 and some classical properties of Turaev’s refined torsion, we can state a result in the spirit of Theorem 2 and Fried’s conjecture. Corollary 6. Let (Z, g) be a compact hyperbolic manifold of odd dimension and denote by ϑ the contact form on S ∗ Z whose Reeb flow is the geodesic flow. Let ρ be the lift to π1 (S ∗ Z) of some acyclic unitary representation π1 (Z) → GL(Cd ). Suppose that η is a contact form path connected to ϑ in the space of contact forms whose Reeb flow are Anosov. Then

(7)

(8)

(9) (−1)q

(10) τ (C • (ϑ, ρ), Γϑ )

(11)

(12) q m(X ,ρ) (−1) η

(13) ζXη ,ρ (s)

(14)

(15) (1 + O(s)) , = |s| τR (ρ)

(16)

(17) (1.4) | {z } τ (C • (η, ρ), Γη )

(18) R-torsion. where (Eρ , ∇ρ ) is the flat vector bundle on M induced by ρ and C • (η, ρ) (resp. C • (ϑ, ρ)) is ∇ ∇ the finite dimensional space of generalized eigenvectors of LXηρ (resp. LXϑρ ) for the resonance 0. This last corollary is non trivial for dim(Z) = 3 since hyperbolic 3-manifolds always admit unitary, acyclic connections by the result of [FN14, section 3]. Let us briefly sketch the proof of Theorem 4, which relies essentially on two variational arguments : we compute the variation of τϑ (∇) when we perturb the contact form ϑ and the connection ∇. As we do so, the space C • (ϑ, ∇) of Pollicott-Ruelle resonant states of L∇ Xϑ for the resonance 0 may radically change. Therefore, it is convenient to consider the space • (ϑ, ∇) instead, which consists of the generalized resonant states for L∇ for resonances C[0,λ] Xϑ.

(19) 6. Y. CHAUBET AND N.V. DANG. s such that |s| ≤ λ, where λ ∈ (0, 1) is chosen so that {|s| = λ} ∩ Res(L∇ Xϑ ) = ∅. Then using [BK07c, Proposition 5.6] and multiplicativity of torsion, one can show that q (λ,∞) • (ϑ, ∇), Γϑ ζXϑ ,ρ (0)(−1) , τϑ (∇) = ±τ C[0,λ] (1.5) (λ,∞). where ζXϑ ,ρ is a renormalized version of ζXϑ ,ρ (we remove all the poles and zeros of ζXϑ ,ρ • (ϑ, ∇), which within {s ∈ C, |s| ≤ λ}), see §5. Thus we can work with the space C[0,λ] behaves nicely under perturbations of X thanks to Bonthonneau’s construction of uniform Anisotropic Sobolev spaces for families of Anosov flows [Bon18], and also under perturbations of ∇. Now consider a smooth family of contact forms (ϑt )t for |t| < ε such that their Reeb vector fields (Xt )t induce Anosov flows. Then Theorem 7 says that for any acyclic ∇, the map t 7→ τϑt (∇) is differentiable and its derivative vanishes. This follows from a result of [BK07c] which allows to compute the variation of the torsion of a finite dimensional complex when the chirality operator is perturbed, and on a variation formula of the map t 7→ ζXt ,ρ (s) for Re(s) big enough obtained in [DGRS18]. Next, consider a smooth family of flat connections z 7→ ∇(z), where z is a complex number varying in a small neighborhood of the origin and write ∇(z) = ∇ + zα + o(z) where α ∈ Ω1 (M, End(E)). Then we show in §7, in the same spirit as the variation formula obtained for the map t 7→ τϑt (∇), that z 7→ τϑ (∇(z)) is complex differentiable and its logarithmic derivative reads −εL∇ X. ∂z |z=0 log τϑ (∇(z)) = −tr[s αKe. ϑ. ,. where ε > 0 is small enough, tr[s is the super flat trace, cf. §3.4, and K : Ω• (M, E) → 0 D • (M, E) is a cochain contraction, that is, it satisfies ∇K + K∇ = IdΩ• (M,E) . On the other hand, we can compute, using the formalism of [DR17b], Z ∇ e −εL−Xe − tr α, ∂z |z=0 log τeϑ ,o (∇(z)) = −tr[s αKe e. e is another cochain contraction, where eϑ is an Euler structure canonically associated to ϑ, K e is a Morse-Smale gradient vector field and e ∈ C1 (M, Z) is a singular one-chain representing X e are cochain contractions, the Euler structure eϑ , cf. §8. Now using the fact that K and K one can see that −εL∇ −εL∇ Xϑ e e X α Ke − Ke = αRε + [∇, αGε ], where Rε is an operator of degree -1 whose kernel is, roughly speaking, the union of graphs of the maps e−εXu , where (Xu )u is a non-degenerate family of vector fields interpolating Xϑ e cf. §8.3, and Gε is some operator of degree -2. Therefore we obtain by cyclicity of and X, the flat trace Z τϑ (∇(z)) [ ∂z |z=0 log = trs αRε − tr α = 0, (1.6) τeϑ ,o (∇(z)) e where the last equality comes from differential topology arguments. Using the analytical structure of the representation variety, we may deduce from (1.6) the first point of Theorem 4. The second point then follows from the invariance of the dynamical torsion under small perturbations of the flow, the fact that τe,o (ρ) = τe0 ,o (ρ)hdet ρ, hi for any other Euler.

(20) DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS. 7. structure e0 , where h ∈ H1 (M, Z) satisfies e = e0 + h (we have that H1 (M, Z) acts freely and transitively on the set of Euler structures, cf. §8), and the fact that, in our notations, eη − eϑ = cs(Xϑ , Xη ) for any other contact form η. Related works. Some analogs of our dynamical torsion were introduced by Burghelea– Haller [BH08b] for vector fields which admit a Lyapunov closed 1–form generalizing previous works by Hutchings [H02], Hutchings–Lee [HL99b, HL99a] dealing with Morse–Novikov flows. In that case, the dynamical torsion depends on a choice of Euler structure and is a partially defined function on Repac (M, d); if d = 1, it is shown in [BH08a] that it extends to a rational map on the Zariski closure of Repac (M, 1) which coincides, up to sign, with Turaev’s refined combinatorial torsion (for the same choice of Euler structure). This follows from previous works of Hutchings–Lee [HL99b, HL99a] who introduced some topological invariant involving circle-valued Morse functions. In both works, the considered object has the form Dynamical zeta function(0) × Correction term where the correction term is the torsion of some finite dimensional complex whose chains are generated by the critical points of the vector field. The chosen Euler structure gives a distinguished basis of the complex and thus a well defined torsion. This is one of the main differences with our work since in the Anosov case, there are no such choices of distinguished currents in C • . However, the chirality operator allows us to overcome this problem as described above. We also would like to mention some interesting related works of Rumin–Seshadri [RS12] where they relate some dynamical zeta function involving the Reeb flow and some analytic contact torsion on 3–dimensional Seifert CR manifolds. Plan of the paper. The paper is organized as follows. In §2, we give some preliminaries about torsion of finite dimensional complexes computed with respect to a chirality operator. In §3, we present our geometrical setting and conventions. In §4, we introduce Pollicott-Ruelle resonances. In §5, we compute the refined torsion of a space of generalized eigenvectors for nonzero resonances and we define the dynamical torsion. In §6, we prove that our torsion is unsensitive to small perturbations of the dynamics. In §7, we compute the variation of our torsion with respect to the connection. In §8, we introduce Euler structures which are some topological tools used to fix ambiguities of the refined torsion. In §9, we introduce the refined combinatorial torsion of Turaev using Morse theory and we compute its variation with respect to the connection. We finally compare it to the dynamical torsion in §10. Acknowledgements. We warmly thank Nalini Anantharaman, Yannick Bonthonneau, Mihajlo Cekić, Alexis Drouot, Semyon Dyatlov, Malo Jézéquel, Thibault Lefeuvre, Julien Marché, Marco Mazzucchelli, Claude Roger, Nicolas Vichery, Jean Yves Welschinger, Steve Zelditch, for asking questions about this work or for interesting discussions related to the paper. Particular thanks are due to Colin Guillarmou who went through the whole paper, helped us correct many errors and is always a source of inspiration. We thank the organizers of the microlocal analysis program in MSRI for the invitation to speak about our result. N.V.D is very grateful to Gabriel Rivière for his friendship, many inspiring maths discussions, his many advices and for the series of works which made the present paper possible..

(21) 8. Y. CHAUBET AND N.V. DANG. Finally, N.V.D acknowledges the incredible patience and love of his wife and daughter, who created the right atmosphere at home which made this possible.. 2. Torsion of finite dimensional complexes We recall the definition of the refined torsion of a finite dimensional acyclic complex computed with respect to a chirality operator, following [BK07c]. Then we compute the variation of the torsion of such a complex when the differential is perturbed. 2.1. The determinant line of a complex. For a non zero complex vector space V , the V determinant line of V is the line defined by det(V ) = dim V V . We declare the determinant line of the trivial vector space {0} to be C. If L is a 1-dimensional vector space, we will denote by L−1 its dual line. Any basis (v1 , . . . , vn ) of V defines a nonzero element v1 ∧ · · · ∧ vn ∈ det(V ). Thus elements of the determinant line of det(V ) should be thought of as equivalence classes of oriented basis of V . Let ∂. ∂. ∂. ∂. ∂. (C • , ∂) : 0 −→ C 0 −→ C 1 −→ · · · −→ C n −→ 0 be a finite dimensional complex, i.e. dim C j < ∞ for all j = 0, . . . , n. We define the determinant line of the complex C • by det(C • ) =. n O. j. det(C j )(−1) .. j=0. Let H • (∂) be the cohomology of (C • , ∂), that is •. H (∂) =. n M. H j (∂),. j=0. H j (∂) =. ker(∂ : C j → C j+1 ) . ran(∂ : C j−1 → C j ). We will say that the complex (C • , ∂) is acyclic if H • (∂) = 0. In that case, det H • (∂) is canonically isomorphic to C. It remains to define the fusion homomorphism that we will later need to define the torsion of a finite dimensional based complex [FT00, §2.3]. For any finite dimensional vector spaces V1 , . . . , Vr , we have a fusion isomorphism µV1 ,...,Vr : det(V1 ) ⊗ · · · ⊗ det(Vr ) → det(V1 ⊕ · · · ⊕ Vr ) defined by µV1 ,...,Vr v11 ∧ · · · ∧ v1m1 ⊗ · · · ⊗ vr1 ∧ · · · ∧ vrmr = v11 ∧ · · · ∧ v1m1 ∧ · · · ∧ vr1 ∧ · · · ∧ vrmr , where mj = dim Vj for j ∈ {1, . . . , r}..

(22) DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS. 9. 2.2. Torsion of finite dimensional acyclic complexes. In the present paper, we want to think of torsion of finite dimensional acyclic complexes as a map ϕC • from the determinant line of the complex to C. We have a canonical isomorphism ∼. ϕC • : det(C • ) −→ C,. (2.1). defined as follows. Fix a decomposition C j = B j ⊕ Aj ,. j = 0, . . . , n,. with B j = ker(∂) ∩ C j and B j = ∂(Aj−1 ) = ∂(C j−1 ) for every j. Then ∂|Aj : Aj → B j+1 is an isomorphism for every j. Fix non zero elements cj ∈ det C j and aj ∈ det Aj for any j. Let ∂(aj ) ∈ det B j+1 denote the image of aj under the isomorphism det Aj → det B j+1 induced by the isomorphism ∂|Aj : Aj → B j+1 . Then for each j = 0, . . . , n, there exists a unique λj ∈ C such that cj = λj µB j ,Aj ∂(aj−1 ) ⊗ aj , where µB j ,Aj is the fusion isomorphism defined in §2.1. Then define the isomorphism ϕC • by ϕ. C•. : c0 ⊗. c−1 1. ⊗ ··· ⊗. n c(−1) n. N (C • ). 7→ (−1). n Y. (−1)j. λj. ∈ C,. j=0. where n. 1X N (C ) = dim Aj dim Aj + (−1)j+1 . 2 •. j=0. One easily shows that ϕC • is independent of the choices of aj [Tur01, Lemma 1.3]. The number τ (C • , c) = ϕC • (c) is called the refined torsion of (C • , ∂) with respect to the element c. The torsion will depend on the choices of cj ∈ det C j . Here the sign convention (that is, • the choice of the prefactor (−1)N (C ) in the definition of ϕC • ) follows Braverman–Kappeler [BK07c, §2] and is consistent with Nicolaescu [Nic03, §1]. This prefactor was introduced by Turaev and differs from [Tur86]. See [Nic03] for the motivation for the choice of sign. 2.3. Torsion with respect to a chirality operator. We saw above that torsion depends on the choice of an element of the determinant line. A way to fix the value of the torsion without choosing an explicit basis is to use a chirality operator as in [BK07c]. Take n = 2r+1 an odd integer and consider a complex (C • , ∂) of length n. We will call a chirality operator an operator Γ : C • → C • such that Γ2 = IdC • , and Γ(C j ) = C n−j ,. j = 0, . . . , n.. Γ induces isomorphisms det(C j ) → det(C n−j ) that we will still denote by Γ. If ` ∈ L is a non zero element of a complex line, we will denote by `−1 ∈ L−1 the unique element such that `−1 (`) = 1. Fix non zero elements cj ∈ det(C j ) for j ∈ {0, . . . , r} and define •. r. (−1) cΓ = (−1)m(C ) c0 ⊗ c−1 ⊗ (Γcr )(−1) 1 ⊗ · · · ⊗ cr. r+1. r. ⊗ (Γcr−1 )(−1) ⊗ · · · ⊗ (Γc0 )−1 ,.

(23) 10. Y. CHAUBET AND N.V. DANG. where r. 1X dim C j dim C j + (−1)r+j . m(C ) = 2 •. j=0. Definition 2.1. The element cΓ is independent of the choices of cj for j ∈ {0, . . . , r}; the refined torsion of (C • , ∂) with respect to Γ is the element τ (C • , Γ) = τ (C • , cΓ ). We also have the following result which is [BK07c, Lemma 4.7] in the acyclic case about the multiplicativity of torsion. ˜ be two acyclic complexes of same length endowed Proposition 2.2. Let (C • , ∂) and (C˜ • , ∂) ˜ Then with two chirality operators Γ and Γ. ˜ = τ (C • , Γ)τ (C˜ • , Γ). ˜ τ (C • ⊕ C˜ • , Γ ⊕ Γ) 2.4. Computation of the torsion with the contact signature operator. Let B = Γ∂ + ∂Γ : C • → C • . B is called the signature operator. Let B+ = Γ∂ and B− = ∂Γ. Denote j C± = C j ∩ ker(B∓ ),. j = 0, . . . , n.. • . Note that B (C j ) ⊂ C n−j−1 , so that B (C j ⊕ C n−j−1 ) ⊂ We have that B± preserves C± + + + + + + j n−j−1 • . If B is invertible, C+ ⊕ C+ . Note that if B is invertible on C • , B+ is invertible on C+ we can compute the refined torsion of (C • , ∂) using the following. Proposition 2.3. [BK07c, Proposition 5.6] Assume that B is invertible. Then (C • , ∂) is acyclic so that det(H • (∂)) is canonically isomorphic to C. Moreover, (−1)r r−1 (−1)j Y r τ (C • , Γ) = (−1)r dim C+ det Γ∂|C+r det Γ∂|C j ⊕C n−j−1 . +. j=0. +. L 2.5. Super traces and determinants. Let V • = pj=0 V j is a graded finite dimensional vector space and A : V • → V • be a degree preserving linear map. We define the super trace and the super determinant of A by trs,V • A =. p X (−1)j trV j A, j=0. dets,V • A =. p Y. j. (detV j A)(−1) .. j=0. We also define the graded trace and the graded determinant of A by trgr,V • A =. p X j=0. (−1)j j trV j A,. detgr,V • A =. p Y. j. (detV j A)(−1) j .. j=0.

(24) DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS. 11. 2.6. Analytic families of differentials. The goal of the present subsection is to give a variation formula for the torsion of a finite dimensional complex when we vary the differential. This formula plays a crucial role in the variation formula of the dynamical torsion, when the representation is perturbed. Indeed, we split the dynamical torsion as the product of the torsion τ (C • (ϑ, ρ), Γϑ ) of some finite dimensional space of Ruelle resonant states and a renormalized value at s = 0 of the dynamical zeta function ζX,ρ (s). Then the following formula allows us to deal with the variation of τ (C • (ϑ, ρ), Γϑ ). Let (C • , ∂) be an acyclic finite dimensional complex of finite odd length n. If S : C • : C • is a linear operator, we will say that it is of degree s if S(C k ) ⊂ C k+s for any k. If S and T are two operators on C • of degrees s et t respectively then the supercommutator of S and T by [S, T ] = ST − (−1)st T S. Cyclicity of the usual trace gives trs,C• [S, T ] = 0 for any S, T . Let U be a neighborhood of the origin in the complex plane and ∂(z), z ∈ U , be a family of acyclic differentials on C • which is complex differentiable at z = 0, that is, ∂(z) = ∂ + za + o(z). (2.2). for some operator a : C • → C • of degree 1. Note that ∂(z) ◦ ∂(z) = 0 implies that the supercommutator [∂, a] = ∂a + a∂ = 0. (2.3) We will denote by C • (z) the complex (C • , ∂(z)). Finally let k : C • → C • be a cochain contraction, that is a linear map of degree 1 such that ∂k + k∂ = IdC • .. (2.4). The existence of such map is ensured by the acyclicity of (C • , ∂). Lemma 2.4. In the above notations, for any chirality operator Γ on C • , the map z 7→ τ (C • (z), Γ) is complex differentiable at z = 0 and

(25) d

(26)

(27) log τ (C • (z), Γ) = −trs,C • (ak). dz

(28) z=0. Note that this implies in particular that trs,C • (ak) does not depend on the chosen cochain contraction k. This is expected since if k 0 is another cochain contraction, [∂, akk 0 ] = ∂akk 0 + akk 0 ∂ = a(k − k 0 ), where by (2.3) and the supertrace of a supercommutator vanishes. Proof. First note that for non zero elements c, c0 ∈ det C • , we have τ (C • (z), c) = [c : c0 ] · τ (C • (z), c0 ), where [c : c0 ] ∈ C satisfies c = [c : c0 ] · c0 . For every j = 0, . . . , n, fix a decomposition C j = Aj ⊕ B j ,. (2.5).

(29) 12. Y. CHAUBET AND N.V. DANG `. where B j = ker ∂ ∩ C j and Aj is any complementary of B j in C j . Fix some basis a1j , . . . , ajj `. of Aj ; then ∂a1j , . . . , ∂ajj is a basis of B j+1 by acyclicity of (C • , ∂). Now let `. `. j−1 cj = a1j ∧ · · · ∧ ajj ∧ ∂a1j−1 ∧ · · · ∧ ∂aj−1 ∈ det C j ,. and n. c = c0 ⊗ (c1 )−1 ⊗ c2 ⊗ · · · ⊗ (cn )(−1) ∈ det C • . Now by definition of the refined torsion, we have for |z| small enough τ (C • (z), c) = ±. n Y. det Aj (z). (−1)j+1. (2.6). j=0. where the sign ± is independent of z and Aj (z) is the matrix sending the basis `. `. j−1 a1j , . . . , ajj , ∂a1j−1 , . . . , ∂aj−1. to the basis `. `. j−1 a1j , . . . , ajj , ∂(z)a1j−1 , . . . , ∂(z)aj−1. (which is indeed a basis of C j for |z| small enough). Let k : C • → C • of degree −1 defined by m k∂am j = aj ,. kam j = 0,. for every j and m ∈ {0, . . . , `j }. Then k∂ + ∂k = IdC • and detAj (z) = det∂B j−1 ⊕B j ∂(z)k ⊕ Id . Now (2.2) and (2.6) imply the desired result, because τ (C • (z), Γ) = [cΓ : c] · τ (C • (z), c) by (2.5). 3. Geometrical setting and notations We introduce here our geometrical conventions and notations. In particular, we adopt the formalism of Harvey–Polking [HP+ 79] which will be convenient to compute flat traces and relate the variation of the Ruelle zeta function with topological objects. 3.1. Twisted cohomology. We consider M an oriented closed connected manifold of odd dimension n = 2r + 1. Let E → M be a flat vector bundle over M of rank d ≥ 1. For k ∈ {0, . . . , n}, we will denote the bundle Λk T ∗ M by Λk for simplicity. We will denote by Ωk (M, E) = C ∞ (M, Λk ⊗ E) the space of E valued k-forms. We set •. Ω (M, E) =. n M. Ωk (M, E).. k=0. Let ∇ be a flat connection on E. We view the connection as a degree 1 operator (as an operator of the graded vector space Ω• (M, E)) ∇ : Ωk (M, E) → Ωk+1 (M, E),. k = 0, . . . , n..

(30) DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS. 13. 2 The flatness of the connection reads ∇ = 0 and thus we obtain a cochain complex • Ω (M, E), ∇. We will assume that the connection ∇ is acyclic, that is, the complex Ω• (M, E), ∇ is acyclic, or equivalently, the cohomology groups n o u ∈ Ωk (M, E) : ∇u = 0 o , k = 0, . . . , n, H k (M, ∇) = n ∇v : v ∈ Ωk−1 (M, E). are trivial. 3.2. Currents and Schwartz kernels. Let 0. D • (M, E) =. n M. 0. D (M, Λk ⊗ E). k=0. E∨. the space of E-valued currents. Let denote the dual bundle of E. We will identify 0k n−k D (M, E) and the topological dual of Ω (M, E ∨ ) via the non degenerate bilinear pairing Z hα, βi = α ∧ β, α ∈ Ωk (M, E), β ∈ Ωn−k (M, E ∨ ), M. where ∧ is the usual wedge product between E-valued forms and E ∨ -valued forms. 0. A continuous linear operator G : Ω• (M, E) → D • (M, E) is called homogeneous if for 0 some p ∈ Z, we have G Ωk (M, E) ⊂ D k+p (M, E) for every k = 0, . . . , n; the number p is called the degree of G and is denoted by deg G. In that case, the Schwartz kernel theorem 0 gives us a twisted current G ∈ D n+p (M × M, π1∗ E ∨ ⊗ π2∗ E) satisfying hGu, viM = hG, π1∗ u ∧ π2∗ viM ×M ,. u ∈ Ωk (M, E),. v ∈ Ωn−k−p (M, E ∨ ),. where π1 and π2 are the projections of M × M onto its first and second factors respectively. 3.3. Integration currents. Let N be an oriented submanifold of M of dimension d, possibly 0 with boundary. The associated integration current [N ] ∈ D n−d (M ) is given by Z. [N ], ω = i∗N ω, ω ∈ Ωd (M ), N. where iN : N → M is the inclusion. We have classically d[N ] = (−1)d+1 [∂N ].. (3.1). For f ∈ Diff(M ), we will set Gr(f ) = {(f (x), x), x ∈ M } the graph of f . Note that Gr(f ) is a n-dimensional submanifold of M × M which is canonically oriented since M is. Therefore, we can consider the integration current over Gr(f ). By definition, we have for any α, β ∈ Ω• (M ) Z. ∗ ∗ [Gr(f )], π1 α ∧ π2 β = f ∗ α ∧ β. M. In particular, [Gr(f )] is the Schwartz kernel of f ∗ : Ω• (M ) → Ω• (M )..

(31) 14. Y. CHAUBET AND N.V. DANG 0. 3.4. Flat traces. Let G : Ω• (M, E) → D • (M, E) be an operator of degree 0. We denote its Schwartz kernel by G and we define . WF0 (G) = (x, y, ξ, η), (x, y, ξ, −η) ∈ WF(G) ⊂ T ∗ (M × M ), where WF denotes the classical H¨ ormander wavefront set, cf [Hör90, §8]. We will also use the notation WF(G) = WF(G) and WF0 (G) = WF0 (G). Assume that WF0 (G) ∩ ∆(T ∗ M ) = ∅,. ∆(T ∗ M ) = {(x, x, ξ, ξ), (x, ξ) ∈ T ∗ M }.. (3.2). Let ι : M → M × M, x 7→ (x, x) be the diagonal inclusion. Then by [Hör90, Theorem 8.2.4] 0 the pull back ι∗ G ∈ D n (M, E ∨ ⊗ E) is well defined and we define the super flat trace of G by tr[s G = htr ι∗ G, 1i, where tr denotes the trace on E ∨ ⊗ E. We will also use the notation tr[gr G = tr[s N G, where N : Ω• (M, E) → Ω• (M, E) is the number operator, that is, N ω = kω for every ω ∈ Ωk (M, E). 0. The notation tr[s is motivated by the following. Let A : C ∞ (M, F ) → D (M, F ) be an operator acting on sections of a vector bundle F . If A satisfies (3.2), we can also define a flat 0 trace tr[ A as in [DZ16, §2.4]. Now if G : Ω• (M, E) → D • (M, E) is an operator of degree 0 0, it gives rise to an operator Gk : C ∞ (M, Fk ) → D (M, Fk ) for each k = 0, . . . , n, where Fk = Λk ⊗ E. Then the link between the two notions of flat trace mentioned above is given by n X tr[s G = (−1)k tr[ Gk . k=0. If Γ ⊂. T ∗M. is a closed conical subset, we let n o 0 0 DΓ• (M, E) = u ∈ D • (M, E), WF(u) ⊂ Γ. (3.3). be the space of E-valued current whose wavefront set is contained in Γ, endowed with its usual topology, cf. [H¨ or90, §8]. If Γ is a closed conical subset of T ∗ (M × M ) not intersecting the conormal to the diagonal N ∗ ∆(T ∗ M ) = {(x, x, ξ, −ξ), (x, ξ) ∈ T ∗ M }, 0. then the flat trace is continuous as a map DΓ• (M × M, π1∗ E ∨ ⊗ π2∗ E) → R. 0. 3.5. Cyclicity of the flat trace. Let G, H : Ω• (M, E) → D • (M, E) be two homogeneous operators. We denote by G, H their respective kernels. If Γ ⊂ T ∗ (M × M ) is a closed conical subset, we define Γ(1) = {(y, η), ∃x ∈ M, (x, y, 0, η) ∈ Γ},. Γ(2) = {(y, η), ∃x ∈ M, (x, y, −η, 0) ∈ Γ}.. Then under the assumption WF(G)(2) ∩ WF(H)(1) = ∅,.

(32) DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS. 15. the operator F = G ◦ H is well defined by [Hör90, Theorem 8.2.14] and its Schwartz kernel F satisfies the wave front set estimate : . WF (F) ⊂ (x, y, ξ, η) | ∃(z, ζ), (x, z, ξ, ζ) ∈ WF0 (G) and (z, y, ζ, η) ∈ WF (H) . If both compositions G ◦ H and H ◦ G are defined, we will denote by [G, H] = G ◦ H − (−1)deg G deg H H ◦ G the graded commutator of G and H. We have the following Proposition 3.1. Let G, H be two homogeneous operators with deg G + deg H = 0 and such that both compositions G ◦ H and H ◦ G are defined and satisfy the bound (3.2). Then we have tr[s [G, H] = 0. The above result follows from the cyclicity of the L2 -trace, the approximation result [DZ16, Lemma 2.8], the relation tr[s [G, H] = tr[ (−1)N F, G , where N is the number operator and tr[ is the flat trace with the convention from [DZ16], see §3.4, and the fact that the map (G, H) 7→ G ◦ H is continuous 0. 0. 0. DΓ• (M × M, π1∗ E ∨ ⊗ π2∗ E) × DΓe• (M × M, π1∗ E ∨ ⊗ π2∗ E) → DΥ• (M × M, π1∗ E ∨ ⊗ π2∗ E) e ⊂ T ∗ (M × M ) such that Γ(2) ∩ Γ e (1) = ∅, and where Υ is for any closed conical subsets Γ, Γ a closed conical subset given in [H¨ or90, 8.2.14]. 3.6. Perturbation of holonomy. Let γ : [0, 1] → M be a smooth curve and α ∈ Ω1 (M, End(E)). ˜ = ∇ + α) along γ|[0,t] . Let Pt (resp. P˜t ) be the parallel transport Eγ(0) → Eγ(t) of ∇ (resp. ∇ Then Z t. P˜t = Pt exp −. P−τ α(γ(τ ˙ ))Pτ dτ. .. (3.4). 0. The above formula will be useful in some occasion. For simplicity, we will denote for any A ∈ C ∞ (M, End(E)) Z Z t A= P−τ A(γ(τ ))Pτ dτ ∈ End Eγ(0) γ. 0. R so that P˜1 = P1 exp − γ α(X) . 4. Pollicott-Ruelle resonances 4.1. Anosov dynamics. Let X be a smooth vector field on M and denote by ϕt its flow. We will assume that X generates an Anosov flow, that is, there exists a splitting of the tangent space Tx M at every x ∈ M Tx M = RX(x) ⊕ Es (x) ⊕ Eu (x),.

(33) 16. Y. CHAUBET AND N.V. DANG. where Eu (x), Es (x) are subspaces of Tx M depending continuously on x and invariant by the flot ϕt , such that for some constants C, ν > 0 and some smooth metric | · | on T M one has |(dϕt )x vs | ≤ Ce−νt |vs |,. t ≥ 0,. |(dϕt )x vu | ≤ Ce−ν|t| |vu |,. t ≤ 0,. vs ∈ Es (x), vu ∈ Eu (x).. We will use the dual decomposition T ∗ M = E0∗ ⊕ Eu∗ ⊕ Es∗ where E0∗ = (Eu ⊕ Es )∗ ,. Es∗ = (E0 ⊕ Es )∗ ,. Eu∗ = (E0 ⊕ Eu )∗ .. (4.1). 4.2. Pollicott-Ruelle resonances. Let ιX denote the interior product with X and • • L∇ X = ∇ιX + ιX ∇ : Ω (M, E) → Ω (M, E). be the Lie derivative along X acting on E-valued forms. Locally, the action of L∇ X is given by 1 the following. Take U a domain of a chart and write ∇ = d + A where A ∈ Ω (M, End(E)). Take w1 , . . . , w` (resp. e1 , . . . , ed ) some local basis of Λk (resp. E) on U . Then for any 1 ≤ i ≤ ` and 1 ≤ j ≤ d, L∇ X (f wi ⊗ ej ) = (Xf )wi ⊗ ej + f (LX wi ) ⊗ ej + f wi ⊗ A(X)ej ,. f ∈ C ∞ (U ),. where LX is the standard Lie derivative acting on forms. In particular, L∇ X is a differential operator of order 1 acting on sections of the bundle Λ• T ∗ M ⊗ E, whose principal part is diagonal and given by X. Denote by Φtk the induced flow on the vector bundle Λk T ∗ M ⊗ E → M , that is, −1. Φtk (β ⊗ v) = T (dϕt )x β ⊗ Pt∇ (x)v,. x ∈ M,. (β, v) ∈ Λk (Tx∗ M ) × Ex ,. t ∈ R,. where Pt∇ (x) is the parallel transport induced by ∇ along the curve {ϕs (x), s ∈ [0, t]}. This induces a map ∇. etLX : Ω• (M, E) → Ω• (M, E). • For Re(s) big enough, the operator L∇ X + s acting on Ω (M, E) is invertible with inverse Z ∞ ∇ ∇ −1 (LX + s) = e−tLX e−st dt. (4.2) 0. The results of [FS11] generalize to the flat bundle case as in [DR17c, §3] and the resolvent −1 0 L∇ , viewed as a family of operators Ω• (M, E) → D • (M, E), admits a meromorphic X +s −1 continuation to s ∈ C with poles of finite multiplicites; we will still denote by L∇ X +s this extension. Those poles are the Pollicott-Ruelle resonances of L∇ X , and we will denote this set by Res(L∇ ). X 4.3. Generalized resonant states. Let s0 ∈ Res(L∇ X ). By [DZ16, Proposition 3.3] we have a Laurent expansion j−1 J(s0 ) ∇ X −1 Πs 0 ∇ j−1 LX + s0 (4.3) LX + s = Ys0 (s) + (−1) j (s − s0 ) j=1.

(34) DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS. where Ys0 (s) is holomorphic near s = s0 , Z −1 1 0 L∇ Πs 0 = ds : Ω• (M, E) → D • (M, E) X +s 2πi Cε (s0 ). 17. (4.4). is an operator of finite rank. Here Cε (s0 ) = {|z − s0 | = ε} with ε > 0 small enough is a small circle around s0 such that Res(L∇ X ) ∩ {|z − s0 | ≤ ε} = {s0 }. Moreover the operators Ys0 (s) and Πs0 extend to continuous operators 0. 0. Ys0 (s), Πs0 : DE•u∗ (M, E) → DE•u∗ (M, E).. (4.5). The space 0. C • (s0 ) = ran(Πs0 ) ⊂ DE•u∗ (M, E) is called the space of generalized resonant states of L∇ X associated to the resonance s0 . 4.4. The twisted Ruelle zeta function. Fix a base point x? ∈ M and identify π1 (M ) with π1 (M, x? ). Let Per(X) be the set of periodic orbits of X. For every γ ∈ Per(X) we fix some base point xγ ∈ Im(γ) and an arbitrary path cγ joining xγ to x? . This path defines an isomorphism ψγ : π1 (M, xγ ) ∼ = π1 (M ) and we can thus define every γ ∈ Per(X) ρ∇ ([γ]) = ρ∇ (ψγ [γ]). The twisted Ruelle zeta function associated to the pair (X, ∇) is defined by Y −s`(γ) ζX,∇ (s) = det Id −ρ∇ ([γ])e , Re(s) > C,. (4.6). γ∈GX. where GX is the set of all primitive closed orbits of X (that is, the closed orbits that generate their class in π1 (M )), `(γ) is the length of the orbit γ and C > 0 is some big constant depending on ρ and X satisfying # , γ ∈ GX. (4.7). | det(I − Pγ )| = (−1)q det(I − Pγ ),. (4.8). kρ∇ ([γ])k ≤ exp(C`(γ)), for some norm k · k on End(Ex? ). For every closed orbit γ, we have. for some q ∈ Z not depending on γ, where Pγ is the linearized Poincaré return map of γ, that is Pγ = dx ϕ−`(γ) |Es (x)⊕Eu (x) for x ∈ Im(γ) (if we choose another point in Im(γ), the map will be conjugated to the first one. This condition is always true when ϕt is contact, in which case we have q = dim Es . Giuletti-Pollicott-Liverani and Dyatlov-Zworski [GLP13, DZ16] showed that ζX,∇ has a meromorphic continuation to C whose poles and zeros are contained in Res(L∇ X ); moreover, 1 ∇ the order of ζX,∇ near a resonance s0 ∈ Res(LX ) is given by n−1 X it follows from [DZ16] that m(s0 ) = (−1)q (−1)k m0k (s0 ), where m0k (s0 ) is the dik=0 mension of Πs0 Ωk (M, E) ∩ ker ιX . We can however repeat the arguments using the identity n−1 n X X det(Id −Pγ ) = − (−1)k k tr Λk dx ϕ−`(γ) instead of the identity det(Id −Pγ ) = (−1)k tr Λk Pγ (see [DZ16,. 1Actually,. k=0. k=0.

(35) 18. Y. CHAUBET AND N.V. DANG. q+1. m(s0 ) = (−1). n X (−1)k kmk (s0 ),. (4.9). k=0. where mk (s0 ) is the rank of the spectral projector Πs0 |Ωk (M,E) . 4.5. Topology of resonant states. Since ∇ commutes with L∇ X , it induces a differential • ∇ on the complexes C (s0 ) for any s0 ∈ Res(LX ). It is shown in [DR17c] that the complexes C • (s0 ), ∇ are acyclic whenever s0 6= 0. Moreover, for s0 = 0, the map Πs0 =0 : Ω• (M, ∇) −→ C • (s0 = 0) is a quasi-isomorphism, that is, it induces isomorphisms at the level of cohomology groups. Since we assumed ∇ to be acyclic, the complex C • (s0 = 0), ∇ is also acyclic. 5. The dynamical torsion of a contact Anosov flow From now on, we will assume that the flow ϕt is contact, that is, there exists a smooth one form ϑ ∈ Ω1 (M ) such that ϑ ∧ (dϑ)r is a volume form on M , ιX ϑ = 1 and ιX dϑ = 0. The purpose of this section is to define the dynamical torsion of the pair (ϑ, ∇). We first introduce a chirality operator Γϑ acting on Ω• (M, E) which is defined thanks to the contact structure. Then the dynamical torsion is a renormalized version of the twisted Ruelle zeta function corrected by the torsion of the finite dimensional space of the generalized resonant states for resonance s0 = 0 computed with respect to Γϑ . This construction was inspired by the work of Braverman-Kappeler on the refined analytic torsion [BK07c]. 5.1. The chirality operator associated to a contact structure. Let VX → M denote the bundle T ∗ M ∩ ker ιX . Note that for k ∈ {0, . . . , n}, we have the decomposition Λk T ∗ M = Λk−1 VX ∧ ϑ ⊕ Λk VX .. (5.1). Indeed, if α ∈ Λk T ∗ M we may write α = (−1)k+1 ιX α ∧ ϑ + α − (−1)k+1 ιX α ∧ ϑ. {z } | {z } | ∈Λk−1 VX ∧ϑ. ∈Λk VX. Let us introduce the Lefschetz map L : Λ • VX u. → Λ•+2 VX 7→ u ∧ dϑ.. Since dϑ is a symplectic form on VX , the maps L r−k induce bundle isomorphisms ∼. L r−k : Λk VX −→ Λ2r−k VX ,. k = 0, . . . , r,. (5.2). see for example [LM87, Theorem 16.3]. Using the above Lefschetz isomorphisms, we are now ready to introduce our chirality operator. k ∗ §2.2]), and study the action of L∇ X on the bundles Λ T M ⊗ E rather than its action on the bundles k ∗ Λ T M ∩ ker ιX ⊗ E, to obtain (4.9)..

(36) DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS. 19. Definition 5.1. The chirality operator associated to the contact form ϑ is the operator Γϑ : Λ• T ∗ M → Λn−• T ∗ M defined by Γ2ϑ = 1 and Γϑ (f ∧ ϑ + g) = L r−k g ∧ ϑ + L r−k+1 f,. f ∈ Λk−1 VX ,. g ∈ Λk VX ,. k ∈ {0, . . . , r}, (5.3). where we used the decomposition (5.1). Note that in particular one has for k ∈ {r + 1, . . . , n}, −1 −1 Γϑ (f ∧ ϑ + g) = L k−r g ∧ ϑ + L k−1−r f. 5.2. The refined torsion of a space of generalized eigenvectors. The operator Γϑ acts also on Ω• (M, E) by acting trivially on E-coefficients. Since LX ϑ = 0, Γϑ and L∇ X commute so that Γϑ induces a chirality operator Γϑ : C • (s0 ) → C n−• (s0 ) • for every s0 ∈ Res(L∇ X ). Recall from §4.5 that the complexes C (s0 ), ∇ are acyclic. The following formula motivates the upcoming definition of the dynamical torsion. Proposition 5.2. Let s0 ∈ Res(L∇ X ) \ {0, 1}. We have τ (C • (s0 ), Γϑ )−1 = (−1)Qs0 detgr,C • (s0 ) L∇ X where Qs0 =. r X. (−1)k (r + 1 − k) dim C k (s0 ). k=0. τ (C • (s. • and 0 ), Γϑ ) ∈ C\0 is the refined torsion of the acyclic complex C (s0 ), ∇ with respect to the chirality Γϑ , cf Definition 2.1. Let us first admit the above proposition; the proof will be given in §§5.5,5.6. 5.3. Spectral cuts. If I ⊂ [0, 1) is an interval, we set X ΠI = Πs0 and CI• = s0 ∈Res(L∇ X) |s0 |∈I. M. C • (s0 ).. s0 ∈Res(L∇ X) |s0 |∈I. • ∇ Note that L∇ X + s acts on C (s0 ) for every s0 ∈ Res(LX ) as −s0 Id +J where J is nilpotent. We thus have for s ∈ / Res(L∇ X) Y (−1)q+1 detgr,CI• L∇ = (s − s0 )m(s0 ) , (5.4) X +s s0 ∈Res(L∇ X) |s0 |∈I. where detgr is the graded determinant, cf. §2.5. Let λ ∈ [0, 1) such that Res(L∇ X ) ∩ {s ∈ C : |s| = λ} = ∅. Now define the meromorphic function (−1)q (λ,∞) • ζX,∇ (s) = ζX,∇ (s)detgr,C[0,λ] L∇ . (5.5) X +s (λ,∞). Then (4.9) and (5.4) show that ζX,∇ has no pole nor zero in {|s| ≤ λ}, so that the number (λ,∞). ζX,∇ (0) is well defined..

(37) 20. Y. CHAUBET AND N.V. DANG. 5.4. Definition of the dynamical torsion. Let 0 < µ < λ < 1 such that for every s0 ∈ Res(L∇ 6 λ, µ. Using Proposition 2.2 and Proposition 5.2 we obtain, X ), one has |s0 | = with notations of §5.3, −1 • • • , Γϑ = (−1)Q(µ,λ] detgr,C(µ,λ] , Γϑ , τ C[0,λ] τ C[0,µ] L∇ X where for an interval I we set QI =. X. Qs0 .. s0 ∈Res(L∇ X) |s0 |∈I. This allows us to give the following Definition 5.3 (Dynamical torsion). The number q (λ,∞) • τϑ (∇) = (−1)Q[0,λ] ζX,∇ (0)(−1) · τ C[0,λ] , Γϑ ∈ C \ 0. (5.6). is independent of the spectral cut λ ∈ (0, 1). We will call this number the dynamical torsion of the pair (ϑ, ∇). Remark 5.4. If cX,∇ sm(0) is the leading term of the Laurent expansion of ζX,∇ (s) at s = 0, then taking λ small enough actually shows that (−1)q τϑ (∇) = (−1)Q0 cX,∇ · τ C • (0), Γϑ . (5.7) In particular, if 0 ∈ / Res(L∇ X ), q. τϑ (∇) = ζX,∇ (0)(−1) .. (5.8). Note that we could have taken (5.7) as a definition of the dynamical torsion; however (5.6) is more convenient to study the regularity of the τϑ (∇) with respect to ϑ and ∇. Remark 5.5. This definition actually makes sense even if ∇ is not acyclic; in that case the formula (5.6) defines an element of the determinant line det H • (M, ∇), under the identifica • , ∇ given by the quasi-isomorphism Π • • tion H • (M, ∇) = H • C[0,λ] [0,λ] : Ω (M, E) → C[0,λ] , cf §4.5. However, we will focus here on the acyclic case. The rest of this section is devoted to the proof of Proposition 5.2. 5.5. Invertibility of the contact signature operator. To prove Proposition 5.2 we shall use §2.4 and introduce the contact signature operator Bϑ = Γϑ ∇ + ∇Γϑ : D0• (M, E) → D0• (M, E), where Γϑ acts trivially on E. We fix in what follows some s0 ∈ Res(L∇ X ) \ {0, 1} and we denote C • (s0 ) by C • for simplicity. We also set C0• = C • ∩ ker(ιX ). The following result will put us in position to apply Proposition 2.3. Lemma 5.6. The operator Bϑ is invertible C • → C • ..

(38) DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS. 21. Proof. We set • Ceven =. M. Ck,. • Codd =. k even. M. Ck.. k odd. • • . Note that because Γ2 = 1, we have Then Bϑ preserves the decomposition C • = Ceven ⊕ Codd ϑ • . Let β ∈ C • • Γϑ . It thus suffices to show that Bϑ is injective on C • = Γϑ Bϑ |Codd Bϑ |Ceven even even such that Bϑ β = 0. Write r X • β= β2k ∈ Ceven , k=0. with β2k = f2k−1 ∧ ϑ + g2k ,. f2k−1 ∈ C02k−1 ,. g2k ∈ C02k ,. k = 0, . . . , r.. Then Bϑ β = 0 writes, since Γϑ ∇(C k ) ⊂ C n−k−1 and ∇Γϑ (C k ) ⊂ C n−k+1 , Γϑ ∇β2k + ∇Γϑ β2(k+1) = 0,. k = 0, . . . , r.. (5.9). Because ∇ does not leave the decomposition (5.1) stable, we need to introduce an operator Ψ : C0• → C0•+1 which mimics the action of ∇. We define Ψµ = ∇µ − (−1)k L∇ X µ ∧ ϑ,. µ ∈ C0k .. Because LX dϑ = 0, the map Ψ satisfies the simple relation Ψ µ ∧ dϑj = (Ψµ) ∧ dϑj , µ ∈ C0• ,. j ∈ N,. (5.10). (5.11). that is, Ψ commutes with L . Also, observe that Ψ2 µ = −L∇ X µ ∧ dϑ,. µ ∈ C0• .. (5.12). Indeed, using the fact that L∇ X and ∇ commute, ∇ k ∇ k+1 L ∇µ − (−1) L µ ∧ ϑ ∧ϑ Ψ2 µ = ∇ ∇µ − (−1)k L∇ µ ∧ ϑ − (−1) X X X k ∇ ∇2 = ∇2 µ + (−1)k+1 ∇ L∇ X µ ∧ ϑ + (−1) LX ∇µ ∧ ϑ − LX µ ∧ ϑ ∧ ϑ = (−1)k+1 (−1)k L∇ X µ ∧ dϑ. Assume first that k ≤ r/2 − 1. Then 2k + 2 ≤ r; we can thus write, with (5.10) in mind, Γϑ ∇β2k = Γϑ ∇f2k−1 ∧ ϑ − f2k−1 ∧ dϑ + ∇g2k ∇ = Γϑ Ψf2k−1 ∧ ϑ − L∇ f ∧ ϑ ∧ ϑ − f ∧ dϑ + Ψg + L g ∧ ϑ 2k−1 2k X 2k−1 X 2k r−2k = Ψf2k−1 + L∇ + Ψg2k − f2k−1 ∧ dϑ ∧ dϑr−2k−1 ∧ ϑ. X g2k ∧ dϑ Similarly we find by (5.11) ∇Γϑ β2k+2 = ∇ f2k+1 ∧ dϑr−2k−1 + g2k+2 ∧ dϑr−2k−2 ∧ ϑ r−2k−1 = Ψf2k+1 − L∇ f ∧ ϑ ∧ dϑ + Ψg ∧ ϑ + g ∧ dϑ ∧ dϑr−2k−2 . 2k+2 2k+2 X 2k+1 (5.13) Thus (5.9) writes, with the decompostion (5.1) in mind, r−2k Ψf2k+1 + g2k+2 ∧ dϑr−2k−1 + Ψf2k−1 + L∇ g =0 (5.14) X 2k ∧ dϑ.

(39) 22. Y. CHAUBET AND N.V. DANG. and . r−2k−2 −L∇ + Ψg2k − f2k−1 ∧ dϑ ∧ dϑr−2k−1 = 0. X f2k+1 ∧ dϑ + Ψg2k+2 ∧ dϑ. (5.15). Then applying Ψ to (5.15) gives, with (5.12) and (5.11), ∇ r−2k−1 r−2k −ΨL∇ − L∇ − Ψf2k−1 ∧ dϑr−2k = 0. X f2k+1 − LX g2k+2 ∧ dϑ X g2k ∧ dϑ −1. ∇ Note that Ψ commutes with L∇ (which exists since s0 6= 0). Then X and thus with LX −1 ∇ applying LX to the above relation we get −1 r−2k −Ψf2k+1 − g2k+2 ∧ dϑr−2k−1 − g2k ∧ dϑr−2k − L∇ Ψf ∧ dϑ = 0. 2k−1 X. Injecting this in (5.14), we obtain ∇ −1 r−2k L∇ − Id g + Id −L Ψf = 0. 2k 2k−1 ∧ dϑ X X Since L r−2k is injective on C02k and L∇ X − Id is invertible (since s0 6= 1), this yields L∇ X g2k + Ψf2k−1 = 0. Applying. −1 L∇ Ψ X. (5.16). to the above equation we get Ψg2k − f2k−1 ∧ dϑ = 0. (5.17). by (5.11); thus (5.15) gives Ψg2k+2 − L∇ f ∧ dϑ ∧ dϑr−2k−2 = 0. X 2k+1 Now repeating this process with k replaced by k − 1 we obtain Ψg2k − L∇ X f2k−1 ∧ dϑ ∧ dϑr−2k = 0. This implies with (5.17) that r−2k+1 Id −L∇ = 0, X f2k−1 ∧ dϑ which leads to f2k−1 = 0 since L r−(2k−1) is injective on C02k−1 and L∇ X − Id is invertible on is invertible. We therefore obtained C • ; thus g2k = 0 by (5.16), since L∇ X β2k = 0,. k ≤ r/2 − 1.. Next assume k ≥ (r + 1)/2. Set k˜ = r − k and ˜ β˜2k+1 = Γϑ β2k ∈ C02k+1 , ˜. ˜ β˜2k−1 = Γϑ β2k+2 ∈ C02k−1 . ˜. Then (5.9) writes Γϑ ∇β˜2k−1 + ∇Γϑ β˜2k+1 = 0. ˜ ˜ Since 2k˜ + 1 ≤ r and we can do exactly as before to get β˜2k−1 = 0 which leads to β2k+2 = 0. ˜ Therefore we obtained β2k = 0, k ≥ (r + 1)/2 + 1. Therefore it remains to show that β2p = 0 and β2(p+1) = 0, where p = br/2c. We will assume that r = 2p + 1 is odd and put p0 = p + 1 (the case r even is similar). Then (5.9) implies, since β2k = 0 for every k 6= p, p0 , ∇Γϑ β2p0 + Γϑ ∇β2p = 0,. Γϑ ∇β2p0 = 0,. ∇Γϑ β2p = 0.. (5.18).

(40) DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS. 23. We can compute, keeping (5.10) in mind, ∇Γϑ β2p0 = ∇ L −1 g2p0 ∧ ϑ + f2p0 −1 −1 = ΨL −1 g2p0 ∧ ϑ + L∇ g2p0 ∧ ϑ ∧ ϑ + L −1 g2p0 ∧ dϑ + Ψf2p0 −1 − L∇ XL X f2p0 −1 ∧ ϑ,. and ∇ Γϑ ∇β2p = Γϑ Ψg2p + L∇ g ∧ ϑ + Ψf ∧ ϑ − L f ∧ ϑ ∧ ϑ − f ∧ dϑ 2p−1 2p−1 X 2p X 2p−1 = Ψg2p ∧ ϑ − f2p−1 ∧ dϑ ∧ ϑ + L∇ X g2p ∧ dϑ + Ψf2p−1 ∧ dϑ. Therefore the first equation of (5.18) implies, since L −1 g2p0 ∧ dϑ = g2p0 , ΨL −1 g2p0 − L∇ X f2p0 −1 − f2p−1 ∧ dϑ + Ψg2p = 0. (5.19). g2p0 + Ψf2p0 −1 + Ψf2p−1 ∧ dϑ + L∇ X g2p ∧ dϑ = 0.. (5.20). and. Applying L∇ X. −1. Ψ to (5.19) leads to −g2p0 − Ψf2p0 −1 − ΨL∇ X. −1. f2p−1 ∧ dϑ + −g2p ∧ dϑ = 0.. Therefore, . Id −L∇ X. −1. . Ψf2p−1 + L∇ − Id g2p ∧ dϑ. X. (5.21). As before this gives Ψf2p−1 + L∇ X g2p = 0 and thus with (5.20) one gets L∇ X g2p + Ψf2p−1 = 0,. g2p0 + Ψf2p0 −1 = 0.. (5.22). Next compute 2 ∇Γϑ β2p = g2p ∧ dϑ2 + Ψf2p−1 ∧ dϑ2 + Ψg2p ∧ ϑ ∧ dϑ − L∇ X f2p+1 ∧ ϑ ∧ dϑ. Therefore the third part of (5.18) gives (we take the ∧ϑ component of the above equation) 2 −L∇ X f2p−1 ∧ dϑ + Ψg2p ∧ dϑ = 0. −1. Applying L∇ Ψ to (5.22) we get Ψg2p = f2p−1 ∧ dϑ; we therefore obtain that f2p−1 = 0 by X injectivity of L 2 on C0r−2 . Thus g2p = 0 by (5.21). Finally compute ∇β2p0 = Ψf2p0 −1 ∧ ϑ + Ψg2p0 + L∇ X g2p0 ∧ ϑ = 0. Therefore the second part of (5.18) implies (since Γϑ ∇β2p0 = 0 is equivalent to ∇β2p0 = 0) Ψf2p0 −1 + L∇ X g2p0 = 0. Therefore by (5.22) we get L∇ X − Id g2p0 = 0, and thus g2p0 = 0. Using (5.19) we conclude that L∇ X f2p0 −1 = 0 which leads to f2p0 −1 = 0..

(41) 24. Y. CHAUBET AND N.V. DANG. 5.6. Proof of Proposition 5.2. We start from Proposition 2.3 which gives us, in view of Lemma 5.6, (−1)r r−1 (−1)j Y r τ (C • , Γϑ ) = (−1)r dim C+ det Γϑ ∇|C+r det Γϑ ∇|C j ⊕C n−j−1 . j=0. +. (5.23). +. where we set as in §2.4 • C+ = C • ∩ ker(∇Γϑ ),. • C− = C • ∩ ker(Γϑ ∇).. We first note that for k ∈ {0, . . . , r} and β ∈ Ωk (M, E), one has ∇Γϑ β = L r−k ∇β − (−1)k ιX ∇β ∧ ϑ + L ιX ∇ιX β − ιX β ∧ ϑ + (−1)k L r−k+1 β − ∇ιX β + (−1)k ιX (β − ∇ιX β) ∧ ϑ , Γϑ ∇β =L r−k−1 ∇β − (−1)k ιX ∇β ∧ ϑ ∧ ϑ + (−1)k L r−k ιX ∇β ,. (5.24). where L j−r = (L r−j |Λj VX )−1 for 0 ≤ j ≤ r. Indeed, using the decomposition (5.1), Γϑ β = (−1)k+1 ιX β ∧ dϑr−k+1 + β + (−1)k ιX β ∧ ϑ ∧ dϑr−k ∧ ϑ = (−1)k+1 ιX β ∧ dϑr−k+1 + β ∧ dϑr−k ∧ ϑ, which leads to ∇Γϑ β = (−1)k+1 ∇ιX β ∧ dϑr−k+1 + ∇β ∧ dϑr−k ∧ ϑ + (−1)k β ∧ dϑr−k+1 k k+1 k+1 r−k+1 k+1 ∇ιX β + (−1) ιX ∇ιX β ∧ ϑ ∧ dϑr−k+1 + (−1) (−1) ιX ∇ιX β ∧ ϑ ∧ dϑ = (−1) + ∇β − (−1)k ιX ∇β ∧ ϑ ∧ dϑr−k ∧ ϑ + (−1)k β + (−1)k ιX β ∧ ϑ ∧ dϑr−k+1 − ιX β ∧ dϑr−k+1 ∧ ϑ, which is exactly the first part of (5.24). The second part follows directly from the decomposition (5.1). Let us introduce, for k ∈ {0, . . . , r}, the operator Jk : C k → C k defined by Jk β = f ∧ ϑ−(−1)k Ψf. (5.25). for any β = f ∧ ϑ + g ∈ C k with f ∈ C0k−1 and g ∈ C0k , and where Ψ is defined in (5.10). k . Indeed, we have for any f ∈ C k−1 and g ∈ C k , Then we claim that Jk takes it values in C+ 0 0 ∇Γϑ (f ∧ ϑ + g) = ∇ g ∧ dϑr−k ∧ ϑ + f ∧ dϑr−k+1 = Ψg ∧ dϑr−k ∧ ϑ + (−1)k g ∧ dϑr−k+1 r−k+1 + Ψf ∧ dϑr−k+1 + (−1)k+1 L∇ ∧ ϑ, X f ∧ dϑ k if and only if which implies that β = f ∧ ϑ + g lies in C+ r−k k Ψg + (−1)k+1 L∇ f ∧ dϑ ∧ dϑ = 0 and Ψf + (−1) g ∧ dϑr−k+1 = 0. X. (5.26).

(42) DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS. 25. But now note that if β = f ∧ ϑ + g = Jk β 0 = f 0 ∧ ϑ − (−1)k Ψf 0 for some β 0 = f 0 ∧ ϑ + g 0 then f = f 0 and g = −(−1)k Ψf , and thus β satisfies the second part of (5.26). We also obtain Ψg = −(−1)k Ψ2 f = −(−1)k L∇ X f ∧ dϑ by (5.12), so the first part of (5.26) is also satisfied. k ; moreover it is obvious that J is a projector. Therefore we can Therefore Jk : C k → C+ k k → C k , we will still denote by J . consider the restricted projection Jk |C k : C+ k + +. The next lemma will be helpful to compute the determinants lying in the product (5.23). k with f ∈ C k−1 and Lemma 5.7. Take k ∈ {0, · · · , r − 1}. Then for any β = f ∧ ϑ + g ∈ C+ 0 g ∈ C0k , one has ∇ ∇ (Γϑ ∇)2 β = L∇ X LX − Id β − LX − Id Jk β.. Proof. Since k < r we can write, thanks to (5.24), Γϑ ∇β = ∇β ∧ ϑ ∧ dϑr−k−1 + (−1)k ιX ∇β ∧ dϑr−k . Therefore ∇Γϑ ∇β = −(−1)k ∇β ∧ dϑr−k + (−1)k ∇ιX ∇β ∧ dϑr−k r−k = (−1)k L∇ X − Id ∇β ∧ dϑ = ιX ∇ιX ∇β − ιX ∇β ∧ ϑ ∧ dϑr−k k + (−1)k (L∇ − Id) ∇β − (−1) ι ∇β ∧ ϑ ∧ dϑr−k , X X ∇ k where we used ∇ιX ∇β = L∇ X ∇β and ιX ∇ιX ∇β = LX ιX ∇β. Since β ∈ C+ one has with (5.24) ∇β − (−1)k ιX ∇β ∧ ϑ ∧ dϑr−k = ιX β − ιX ∇ιX β ∧ dϑr−k+1 .. This leads to ∇Γϑ ∇β = ιX ∇ιX ∇β − ιX ∇β ∧ ϑ ∧ dϑr−k ι β − ι ∇ι β ∧ dϑr−k+1 . + (−1)k L∇ − Id X X X X ∇ Since ιX ∇ιX ∇β − ιX ∇β = L∇ X − Id ιX ∇β and ιX β − ιX ∇ιX β = Id −LX ιX β, we obtain r−k ∇ r−k+1 ∇Γϑ ∇β = L∇ + (−1)k L∇ , X − Id ιX ∇β ∧ ϑ ∧ dϑ X − Id Id −LX ιX β ∧ dϑ and thus by definition of Γϑ 2 ∇ Γϑ ∇Γϑ ∇β = −(−1)k Id −L∇ X ιX β ∧ ϑ + LX − Id ιX ∇β.. (5.27). Now, writing β = f ∧ ϑ + g where ιX f = 0 and ιX g = 0, we have ∇β = ∇f ∧ ϑ − (−1)k f ∧ dϑ + ∇g, k ∇ ιX ∇β = L∇ X f ∧ ϑ + (−1) ∇f + LX g,. ιX β ∧ ϑ = −(−1)k f ∧ ϑ.. (5.28).